MinT - Best Known (188−28, 188, s)-Nets in Base 2

Best Known (188−28, 188, s)-Nets in Base 2

(188−28, 188, 624)-Net over F₂ — Constructive and digital

Digital (160, 188, 624)-net over F₂, using

2² times duplication [i] based on digital (158, 186, 624)-net over F₂, using
- trace code for nets [i] based on digital (3, 31, 104)-net over F₆₄, using
  - net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

(188−28, 188, 2056)-Net over F₂ — Digital

Digital (160, 188, 2056)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹⁸⁸, 2056, F₂, 4, 28) (dual of [(2056, 4), 8036, 29]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2¹⁸⁸, 8224, F₂, 28) (dual of [8224, 8036, 29]-code), using
  - 1 times truncation [i] based on linear OA(2¹⁸⁹, 8225, F₂, 29) (dual of [8225, 8036, 30]-code), using
    - constructionÂ X applied to C([0,14]) âŠ‚ C([0,12]) [i] based on
      1. linear OA(2¹⁸³, 8193, F₂, 29) (dual of [8193, 8010, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 8193Â | 2²⁶âˆ’1, defining interval IÂ =Â [0,14], and minimum distance dÂ â‰¥ |{−14,−13,…,14}|+1Â =Â 30 (BCH-bound) [i]
      2. linear OA(2¹⁵⁷, 8193, F₂, 25) (dual of [8193, 8036, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 8193Â | 2²⁶âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]
      3. linear OA(2⁶, 32, F₂, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
        Reedâ€“Muller code RM(1,5) [i]
        caps in base bÂ =Â 2 [i]

(188−28, 188, 66638)-Net in Base 2 — Upper bound on s

There is no (160, 188, 66639)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 392 337764 073104 215140 314167 216575 887960 390356 624079 860261 > 2¹⁸⁸ [i]